# On Trivial Extensions and Higher Preprojective Algebras

## TL;DR

This paper establishes a deep connection between twisted trivial extensions and higher preprojective algebras for Koszul n-homogeneous algebras, extending classical correspondences in noncommutative algebra.

## Contribution

It demonstrates that the quadratic dual of certain twisted trivial extensions is isomorphic to higher preprojective algebras, providing a new perspective on algebraic dualities.

## Key findings

- Quadratic dual of twisted trivial extension equals higher preprojective algebra
- Application to τ-slice algebras of stable n-translation algebras
- Noncommutative Bernstein-Gelfand-Gelfand correspondence

## Abstract

In this paper, we show that for a Koszul $n$-homogeneous algebra $\Lambda$, the quadratic dual of certain twisted trivial extension is the $(n+1)$-preprojective algebra of its quadratic dual, that is, $ (\Delta_{\nu}\Lambda)^{!,op} \simeq\Pi( \Lambda^{ !, op })$. This is applied to the $\tau$-slice algebras of stable $n$-translation algebras and gives a noncommutative version of Bernstein-Gelfand-Gelfand correspondence for such algebras.

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Source: https://tomesphere.com/paper/1902.04772